Finite length and trapped-particle diocotron modes

Diocotron modes are discussed for a finite length nonneutral plasma column under the assumption of bounce averaged E × B drift dynamics and small Debye length. In this regime, Debye shielding forces the mode potential to be constant along field lines within the plasma. One can think of the plasma as a collection of magnetic field aligned rods that undergo E × B drift across the field and adjust their length so as to maintain the condition ∂δ&phis;/∂ z = 0 inside the plasma. Using a Green function to relate the perturbed charge density and the perturbed potential, imposing the constraint ∂δ&phis;/∂ z = 0, and discretizing yields a matrix eigenvalue problem. The solutions include the full continuum and discrete stable and unstable diocotron modes. Finite column length introduces a new set of discrete diocotron-like modes. Also, finite column length makes possible the exponential growth of l = 1 diocotron modes, long observed in experiments. The model is extended to include the dependence of a particle's bounce averaged rotation frequency on its axial energy. For certain distributions of axial energies, this dependence can substantially affect the instability.; Recent experiments have characterized trapped-particle modes on a non-neutral plasma column[17]. Theoretical predictions for the mode frequency, damping rate, and eigenmode structure are developed here. The modes are excited on a nonneutral plasma column in which classes of trapped and passing particles have been created by the application of a potential barrier. The barrier is created by applying a voltage to an azimuthally symmetric section of the wall near the axial mid-point of the column. Low energy particles near the edge of the column are trapped in one end or the other, while high energy particles near the center of the column transit the entire length. The modes have azimuthal variation l = 1, 2, … and odd z-symmetry. The trapped particles on either side of the barrier execute E × B drift oscillations producing density perturbations that are 180° out of phase with each other, while passing particles run back and forth along the field lines attempting to Debye shield the perturbed charge density. The mode is damped by collisional scattering across the separatrix between trapped and passing particles. The damping rate is calculated using a boundary layer analysis of the Fokker-Planck equation. It is also shown that the damping is associated with radial transport of plasma particles.